Robust Sliding Mode ControlUsing Fuzzy Controller · 2020. 4. 1. · as any real time system such as magnetic levitation system. We choose the sliding surface as = + 3 (13) This sliding

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438

Volume 4 Issue 3, March 2015

www.ijsr.net Licensed Under Creative Commons Attribution CC BY

Robust Sliding Mode ControlUsing Fuzzy

Controller

1Sonu Kumar,

2Dr. M. J. Nigam

1, 2IIT Roorkee, Electronics & Communication, Roorkee, Uttarakhand, India

Abstract: In this paper, a fuzzy sliding mode controller is designed for a nonlinear control system. Instead of fuzzifying the error and

the error change, we fuzzify the sliding surface. Sliding surface is formed from error and error change, i.e. we are converting a two

variable problem into a single variable problem. Now, the basis of rules formation is the sliding surface and the output is deduced by the

proper organization of inference. This fuzzy sliding mode controller is based on Variable Structure Control (VSC) theory that introduces

the boundary layer in the switching surface and also the control law is continuously approximated in this layer which guarantees the

stability and robustness. By using Lyapunov’s theory, we ensure the stability of this controller.

Keywords: Sliding mode control, fuzzy logic controller, Lyapunov Theory, variable structure control, robust control.

1. Introduction

Sometimes we choose a simplified representation for the

dynamics of a nonlinear system for ease of mathematics and

usually because of some unknown plant parameters, there

exists model imprecision. This affects the nonlinear control

system in a very drastic manner. Robust control is one of the

most effective techniques to deal with these model

uncertainties.

To control complex systems or imperfectly modeled systems

using fuzzy logic, a lot of efforts have been made in the past

and these have been fruitful in many areas. However the

designing of fuzzy logic controller greatly depends upon the

expert’s knowledge or trial and errors. Furthermore, fuzzy

controller does not guarantee the stability and the robustness

due to the linguistic expressions of the fuzzy control [4].

Sliding mode is an important robust control approach. It is a

distinct type of Variable Structure Control (VSC) in which

the control law is continuously changed during the control

process according to some well-defined rule which depends

on the states of the system [5]. Here the control law is a

switching control. Sliding mode control will force the state

trajectory of the system to reach, and subsequently remain on

the predetermined surface. After reaching sliding surface, the

system becomes completely inconsiderate to parameter

variations and external disturbances [3]. So, robustness is

achieved but this method uses drastic changes of the control

input which produces the phenomenon of chattering. The

chattering may excite the high frequency components of the

system that is neglected while modelling. In order to avoid

the problem of chattering, the boundary layer is introduced

and the control input is approximated in this boundary layer.

So, the fuzzy controller is designed to deal with this

chattering problem. In many fuzzy control systems, the rules

are generated by using the error and the error change, but

here we fuzzify the sliding surface i.e. ultimately we are

reducing the order of the system [2]. The sliding surface,

denoted by s, is a combination of the error and the change in

error. This controller can get rid of the chattering problem

and guarantees the stability and the robustness.

2. Sliding Mode Control

Let the nth order system as follows

𝑥 𝑛 𝑡 = 𝑓 𝑥, 𝑡 + 𝑏 𝑥, 𝑡 𝑢 𝑡 + 𝑑(𝑥, 𝑡) (1)

where𝑥𝑇 = [𝑥 𝑥 … . . 𝑥(𝑛−1)] is the state matrixof the system,

𝑏(𝑥, 𝑡) and 𝑓(𝑥, 𝑡) are the functions which are determining

the dynamics of the system, 𝑑(𝑥, 𝑡) is the disturbance, and

𝑢(𝑡) is the control input signal.

Defining the error as

𝑒 𝑡 = 𝑥 𝑡 − 𝑥𝑜(𝑡) (2)

where𝑥𝑜(𝑡) is the desirable state. Now, defining the sliding

surface for the system as

𝑆 𝑥, 𝑡 = 𝑒 𝑡 + 𝜆𝑒(𝑡) (3)

which is converting a higher order nonlinear function into a

first order linear functionwhich consists error and the error

change with a slope of 𝛌. The trajectory of the system is kept

onto this sliding surface with the help of a control law. And

this control law can be found out by Lyapunov’s direct

method.

Assume Lyapunov’s function as

𝑉 𝑠 =1

2𝑆2(𝑥, 𝑡) (4)

where𝑉(𝑠) is a positive definite function for ∀ 𝑆 𝑥, 𝑡 > 0.

The stability condition can be given as

𝑉 𝑠 = 𝑆𝑆 ≤ −𝜂│𝑆(𝑥, 𝑡)│ (5)

where𝜂 is a strictly positive real constant.On substituting

equation (1) into equation (5) and after doing some

simplifications, wecan get,

𝑆 𝑓 + 𝑏 + 𝑢 + 𝑑 − 𝑥 𝑜 + 𝜆𝑒 ≤ −𝜂│𝑆│ (6)

So, the reaching condition can be satisfied by choosing a

control input as

𝑢 =−𝑓+𝑥 𝑜−𝜆𝑒

𝑏− 𝐾𝑠𝑔𝑛(𝑆) (7)

K is a bounded strictly positive real constant which is

depending on the external disturbances and the estimated

system parameters. The 𝑠𝑔𝑛() function denotes the signum

function which is a sensing function which senses the

condition at which the trajectory meets with the sliding

surface. 𝑠𝑔𝑛()can be given as

𝑠𝑔𝑛 𝑠 =

1 𝑓𝑜𝑟 𝑠 > 00 𝑓𝑜𝑟 𝑠 = 0

−1 𝑓𝑜𝑟 𝑠 < 0

(8)

Paper ID: SUB152442 1470





3. Chattering

From (7) we can see that the sign of control input depends on

the sign of 𝑠𝑔𝑛(𝑆). The sign of the control input will change

as the trajectory crosses the line 𝑆 = 0. This will produce a

zig-zag motion of the trajectory which is referred to as

chattering. As all physical systems are low frequency

systems in nature, so we neglect the high frequency

components of the systems for the ease of

modeling.Chattering generates the high frequency signals

into the system which will affect the performance of the

system. To avoid chattering, we will design a fuzzy sliding

surface through fuzzy logic controller.

Figure 1: Sliding mode control with chattering

4. Fuzzy Sliding Mode Controller

As mentioned before, we are not fuzzifying the error and the

error change as done in conventional fuzzy controller design.

Instead, we are fuzzifying the sliding surface which is the

combination of the error and the error change. Now, the basis

of the rule formation is the sliding surface which ultimately

reduces the number of fuzzy rules also.

Firstly, the crisp sliding surface 𝑠 = 0is extended to the

fuzzy sliding surface defined by the expression for the fuzzy

controller design as,

𝑠 𝑖𝑠 𝑍𝐸𝑅𝑂 (9)

where 𝑠 : linguistic variable for s and

ZERO: one of the fuzzy set of fuzzy sliding surface.

The following fuzzy sets are introduced in the universe of

discourse of s as

𝑈(𝑠 ) = {𝑁𝑀𝐵,𝑁𝐵,𝑁𝑀,𝑍𝑅,𝑃𝑀,𝑃𝐵 ,𝑃𝑀𝐵} (10)

Similarly, the control input is partitioned as

𝑈 𝑢 = {𝑆𝑀𝐴𝐿𝐿𝐸𝑆𝑇, 𝑆𝑀𝐴𝐿𝐿𝐸𝑅, 𝑆𝑀𝐴𝐿𝐿, 𝑀𝐸𝐷𝐼𝑈𝑀 ,𝐵𝐼𝐺,𝐵𝐼𝐺𝐺𝐸𝑅 ,𝐵𝐼𝐺𝐺𝐸𝑆𝑇}(11)

Fuzzy rules are as follows:

RULE 1: If s is NMB then u is BIGGEST

RULE 2: If s is NB then u is BIGGER

RULE 3: If s is NM then u is BIG

RULE 4: If s is ZR then u is MEDIUM

RULE 5: If s is PM then u is SMALL

RULE 6: If s is PB then u is SMALLER

RULE 7: If s is PMB then u is SMALLEST

The membership functions of the above fuzzy sets are shown

in Figure 2(a) and 2(b):

Figure 2(a): Membership functions for the sliding surface, s

Figure 2(b): Membership functions for control input, u

The result of inference from these seven rules is shown in

Figure 3.

Figure 3: The result of the inference

5. Simulation and Results

The simulations are performed on a nonlinear control system

having the transfer function as follows:

𝐺 𝑠 =2502 .96

𝑠2−981 (12)

which is an unstable system having its poles on the right side

of the s-plane. The above transfer function can be regarded

as any real time system such as magnetic levitation system.

We choose the sliding surface as

𝑆 = 𝑒 + 3𝑒 (13)

This sliding surface is given to fuzzy controller where the

fuzzy rules are made. Rules are based on the sliding surface.

Then control input is generated as follows:

𝑢 = 𝑥 𝑑 + 3𝑒 − 99𝑥 + 0.2𝑠𝑔𝑛(𝑒 + 3𝑒) (14)

The Simulink model and simulation results are shown in

following figures:






Figure 4: Simulink model for the system

Figure 5: Fuzzy sliding mode controller

Figure 6: Phase plane trajectory of the system

Figure 7: Convergences of state trajectories

Figure 8: Sliding mode control law, u

Simulation is done on MATLAB using Simulink to verify

the controller. The results are shown in the figure 6, figure 7,

and figure 8. From figure 6, we can see phase plane

trajectory plot starts from the initial point (2, 0), move

towards the sliding surface. After reaching the sliding

surface, trajectory slides along it to reach equilibrium point

(0, 0). From figure 7, we can see both x and y reach 0 after

4.5 seconds approximately. Figure 8 shows the sliding mode

control law, u.

6. Conclusion

This paper highlights the basic idea of sliding mode control,

chattering phenomenon and methods to avoid chattering. We

designed a fuzzy controller in which sliding surface is

fuzzified which eliminates the chattering problem. This

compresses the two variables problem into a single variable

problem i.e. converting a second order nonlinear system into

a first order linear system. By this we are also able to reduce

the no. of rules in fuzzy controller. Then by applying

Lyapunov’s theory, we guarantee the stability and

robustness.

References

[1] ChintuGurbani and Dr. Vijay Kumar, “Designing Robust

Control by Sliding Mode Control Technique,” Advance

in Electronic and Electric Engineering ISSN 2231-1297,

Volume 3, Number 2, pp. 137-144, 2013.

[2] Sung-Woo Kim and Ju-Jang Lee, “Design of a Fuzzy

Controller with Fuzzy Sliding Surface,” Fuzzy Sets and

Systems 71, Elsevier Science B.V., pp. 359-367, 1995.

[3] Sung-Woo Kim and Ju-Jang Lee, “Fuzzy Logic Based

Sliding Mode Control,” Fifth IFSA World Congress, pp.

822-825, 1993.

[4] D. Dubois and H. Prade, “Fuzzy Sets and Systems:

Theory and Applications” (Academic Press, Orlando, FL,

1980).

[5] Sung-Woo Kim and Ju-Jang Lee, “Variable structure

system and fuzzy sliding surface,” Trans. KIEE, 42(1993)

87-96 (in Korean).

[6] R. Palm, “Sliding Mode Fuzzy Control,” IEEE Int. Conf.

on Fuzzy Systems, pp. 519-526, 1992

Author Profile

Sonu Kumar received the B.Tech. degree in

Instrumentation and Control Engineering from Bharati

Vidyapeeth’s College Of Engineering in 2012. Then he

received the M.Tech. degree in System Modeling and

Control from IIT Roorkee in 2015.

Dr. M. J. Nigam received the B.Tech. degree in

Electronics and Communication Engineering from

REC- Warangal (A.P.) in 1976. Then he received the

M.E. and Ph.D. degree in Electronics and

Communication Engineering from IIT Roorkee (formerly

University of Roorkee) in 1978 and 1992 respectively. His areas of

interest are digital image processing, guidance and control in

navigational systems including high resolution rate.


Documents

Robust Sliding Mode ControlUsing Fuzzy Controller · 2020. 4. 1. · as any real time system such as magnetic levitation system. We choose the sliding surface as = + 3 (13) This sliding